umm

i think what you just described is texture mapping.
Texture mapping is when you take a 2D image and apply it to a 3d mesh.

bump mapping is when you take a 2d image that is in greyscale, apply it to a 3d mesh, and the lighter parts of that 2d image gives the impression that the surface is raised. so light areas in the 2d image when applied to the 3d mesh will look like it's a bump

normal mapping has a similar or same (not sure) effect as bump mapping. the thing with normal mapping is you can make a normal map from high res 3d meshes and make a 2d picture out of it using the UVs and then you can apply it to a low res mesh like you would with a bump map. A normal map isn't greyscale, it has three channels to it.

both bump mapping and normal mapping give the impression that the mesh has more geometry than it really has

i'm still unsure how to explain normal mapping myself but this is what i understand of it...
if anyone knows how to explain this better then go ahead

btw, in terms of 3d, a normal is the direction that a surface is facing